Guide to Measurement and Instrumentation -- Measurement Uncertainty

1 Introduction

2 Sources of Systematic Error

• Measurements in electric circuits

3 Reduction of Systematic Errors

4 Quantification of Systematic Errors

• Environmental condition errors
• Calibration errors
• System disturbance errors
• Measurement system loading errors

5 Sources and Treatment of Random Errors

6 Statistical Analysis of Measurements Subject to Random Errors

7 Aggregation of Measurement System Errors

• Error in a sum
• Error in a difference
• Error in a product
• Error in a quotient

8 Summary

9 Quiz / Problems

Introduction

We have already been introduced to the subject of measurement uncertainty in the last section, in the context of defining the accuracy characteristic of a measuring instrument. The existence of measurement uncertainty means that we would be entirely wrong to assume (although the uninitiated might assume this) that the output of a measuring instrument or larger measurement system gives the exact value of the measured quantity. Measurement errors are impossible to avoid, although we can minimize their magnitude by good measurement system design accompanied by appropriate analysis and processing of measurement data.

We can divide errors in measurement systems into those that arise during the measurement process and those that arise due to later corruption of the measurement signal by induced noise during transfer of the signal from the point of measurement to some other point. This section considers only the first of these, with discussion on induced noise being deferred to Section 6.

It’s extremely important in any measurement system to reduce errors to the minimum possible level and then to quantify the maximum remaining error that may exist in any instrument output reading. However, in many cases, there is a further complication that the final output from a measurement system is calculated by combining together two or more measurements of separate physical variables. In this case, special consideration must also be given to determining how the calculated error levels in each separate measurement should be combined to give the best estimate of the most likely error magnitude in the calculated output quantity. This subject is considered in Section 3.7.

The starting point in the quest to reduce the incidence of errors arising during the measurement process is to carry out a detailed analysis of all error sources in the system. Each of these error sources can then be considered in turn, looking for ways of eliminating or at least reducing the magnitude of errors. Errors arising during the measurement process can be divided into two groups, known as systematic errors and random errors.

Systematic errors describe errors in the output readings of a measurement system that are consistently on one side of the correct reading, that is, either all errors are positive or are all negative. (Some books use alternative name bias errors for systematic errors, although this is not entirely incorrect, as systematic errors include errors such as sensitivity drift that are not biases.) Two major sources of systematic errors are system disturbance during measurement and the effect of environmental changes (sometimes known as modifying inputs), as discussed. Other sources of systematic error include bent meter needles, use of uncalibrated instruments, drift in instrument characteristics, and poor cabling practices.

Even when systematic errors due to these factors have been reduced or eliminated, some errors remain that are inherent in the manufacture of an instrument. These are quantified by the accuracy value quoted in the published specifications contained in the instrument data sheet.

Random errors, which are also called precision errors in some books, are perturbations of the measurement either side of the true value caused by random and unpredictable effects, such that positive errors and negative errors occur in approximately equal numbers for a series of measurements made of the same quantity. Such perturbations are mainly small, but large perturbations occur from time to time, again unpredictably. Random errors often arise when measurements are taken by human observation of an analogue meter, especially where this involves interpolation between scale points. Electrical noise can also be a source of random errors. To a large extent, random errors can be overcome by taking the same measurement a number of times and extracting a value by averaging or other statistical techniques, as discussed in Section 3.5. However, any quantification of the measurement value and statement of error bounds remains a statistical quantity. Because of the nature of random errors and the fact that large perturbations in the measured quantity occur from time to time, the best that we can do is to express measurements in probabilistic terms: we may be able to assign a 95 or even 99% confidence level that the measurement is a certain value within error bounds of, say, _1%, but we can never attach a 100% probability to measurement values that are subject to random errors. In other words, even if we say that the maximum error is __0:5 of the measurement reading, there is still a 1% chance that the error is greater than _0:5 .

Finally, a word must be said about the distinction between systematic and random errors. Error sources in the measurement system must be examined carefully to determine what type of error is present, systematic or random, and to apply the appropriate treatment. In the case of manual data measurements, a human observer may make a different observation at each attempt, but it’s often reasonable to assume that the errors are random and that the mean of these readings is likely to be close to the correct value. However, this is only true as long as the human observer is not introducing a parallax-induced systematic error as well by persistently reading the position of a needle against the scale of an analogue meter from one side rather than from directly above. A human-induced systematic error is also introduced if an instrument with a first-order characteristic is read before it has settled to its final reading.

Wherever a systematic error exists alongside random errors, correction has to be made for the systematic error in the measurements before statistical techniques are applied to reduce the effect of random errors.

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